Wild West's so far successfully replaced doezens of third world regimes. It is fair to say that methodology had been tested and perfected and it works like clockwork. It is more a question of uniting political will and choosing good timing.
There are many functional democracies in Muslim world, like Turkey, Egypt, Iran and Iraq. Obviously they are far from standards of European democracies, but with exception of Iran, none of them is exporting religious wars.
Democratic Saudi Arabia will be much more open country, it would attract even more foreign investment and it would even more focus on its internal problems like high unemployment, instead of waging Jihads and torturing it's citizens.
Recent overthrows in the Muslim world have had disastrous consequences. The stupid overthrow of the Iranian leader Mosaddegh in the 1950s led to a friendly government for only twenty years or so, before it resulted in a violent revolution installing Khomeini's hostile theocracy.
Other regime changes, even though they have been disasters, have been directed at easy targets. Saudi Arabia is not an easy target, having a large, well-funded, and well-equipped military with modern western equipment and a spectacularly large wealth on which to draw. Any attempt at regime change in Saudi Arabia would lead to a global cataclysm.
Your examples of "functioning democracies" really stretch any reasonable definition of the term. Turkey was founded as a secular state. Its current leader, Erdogan, is an Islamist and has made Turkey increasingly Islamist. As it has become more Islamist, so it has become less democratic. Its people voted recently in a referendum to make Erdogan essentially an autocrat. Erdogan himself said, several years ago, "Democracy is like a tram. You ride it until you arrive at your destination, then you step off.”
Egypt had more-or-less fair and open elections in 2012 which installed the rabidly Islamist Muslim Brotherhood in power in the country. The situation was so unpalatable that the Egyptian military took over in a coup and massacred hundreds of protesters doing so. The poll was re-run a year later with military-approved candidates in the running. The Muslim Brotherhood boycotted the election.
Iraq's politics are a mess. It faces a massive perennial Islamist insurrection and continual armed conflict. It could not stably exist without US military intervention. What is the cumulative and ongoing cost in lives of Iraq's "democracy"?
The Economist magazine's Intelligence Unit ranks Iran at 150th (out of 167) in the globe in its Democracy Index in 2017, only nine places ahead of Saudi Arabia, whom you say requires a regime change to make it democratic. It has an unelected supreme leader, and its "elections" are not free and fair, with ballot-stuffing by the regime to ensure that their preferred candidate is the "winner". The Economist divides nations into one of four types: Full democracy, flawed democracy, hybrid regime and authoritarian regime. Iran is classed as an authoritarian regime.
You share the regime changers' delusional assumption, proven false again and again, that simply granting a country a democracy makes it stable, law-governed and able to join the international ranks of first world countries. It does not. Democracy does not give a country an ordered liberty like what exists in the West. Instead it is an ordered liberty that makes a democracy viable. Trying to prescribe democracy from outside does not and cannot work.
The only aim that unites the peoples of most middle eastern nations is the destruction of Israel. Parties which have this aim frequently poll very well in the region, when elections are granted. Democracy does not equal less Islamism, it equals more Islamism.